Maximum Throughput of Multiple Access Channels in Adversarial Environments

TL;DR

This paper introduces a new algorithm for deterministic distributed broadcasting in adversarial environments that achieves maximum throughput and analyzes the limitations of fairness and stability in small systems.

Contribution

The paper develops an optimal throughput algorithm for multiple access channels under adversarial injection and proves fundamental limitations on fairness and stability.

Findings

Achieves throughput 1 against leaky-bucket adversaries.

Proves the queue size upper bound is tight.

Identifies stability and fairness limitations in small systems.

Abstract

We consider deterministic distributed broadcasting on multiple access channels in the framework of adversarial queuing. Packets are injected dynamically by an adversary that is constrained by the injection rate and the number of packets that may be injected simultaneously; the latter we call burstiness. The maximum injection rate that an algorithm can handle in a stable manner is called the throughput of the algorithm. We develop an algorithm that achieves throughput $1$ for any number of stations against leaky-bucket adversaries. The algorithm has $O(n^2+\text{burstiness})$ packets queued simultaneously at any time, where $n$ is the number of stations; this upper bound is proved to be best possible. An algorithm is called fair when each packet is eventually broadcast. We show that no algorithm can be both stable and fair for a system of at least two stations against leaky-bucket adversaries. We study in detail small systems of exactly two and three stations against window adversaries to exhibit differences in quality of broadcast among classes of algorithms. For two stations, we show that fair latency can be achieved by a full sensing algorithm, while there is no stable acknowledgment based algorithm. For three stations, we show that fair latency can be achieved by a general algorithm, while no full sensing algorithm can be stable. Finally, we show that algorithms that either are fair or do not have the queue sizes affect the order of transmissions cannot be stable in systems of at least four stations against window adversaries.